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Scientific Reports

Quantization of geometric phase with integer and fractional topological characterization in a quantum Ising chain with long-range interaction

Sujit Sarkar

Published Apr 11, 2018• 24 citations
Publisher

Abstract

An attempt is made to study and understand the behavior of quantization of geometric phase of a quantum Ising chain with long range interaction. We show the existence of integer and fractional topological characterization for this model Hamiltonian with different quantization condition and also the different quantized value of geometric phase. The quantum critical lines behave differently from the perspective of topological characterization. The results of duality and its relation to the topological quantization is presented here. The symmetry study for this model Hamiltonian is also presented. Our results indicate that the Zak phase is not the proper physical parameter to describe the topological characterization of system with long range interaction. We also present quite a few exact solutions with physical explanation. Finally we present the relation between duality, symmetry and topological characterization. Our work provides a new perspective on topological quantization.

DOI
10.1038/s41598-018-24136-1

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An attempt is made to study and understand the behavior of quantization of geometric phase of a quantum Ising chain with long range interaction. We show the existence of integer and fractional topological characterization for this model Hamiltonian with different quantization condition and also the different quantized value of geometric phase. The quantum critical lines behave differently from the perspective of topological characterization. The results of duality and its relation to the topological quantization is presented here. The symmetry study for this model Hamiltonian is also presented. Our results indicate that the Zak phase is not the proper physical parameter to describe the topological characterization of system with long range interaction. We also present quite a few exact solutions with physical explanation. Finally we present the relation between duality, symmetry and topological characterization. Our work provides a new perspective on topological quantization.
Abstract Study of symmetry, topology and geometric phase can reveal many new and interesting results on the topological states of matter. Here we present a completely new and interesting result of symmetry, topology and quantization of geometric phase along with the physical explanation for two different symmetry classes. We present a detailed study of the auxiliary space for two different symmetry classes of Hamiltonians. We show explicitly that the origin of the auxiliary space inside the curve is only a necessary condition but it is not a sufficient condition for the topological state. One of the most interesting results is that same symmetry-class Hamiltonians show different behavior in topology and quantized geometric phase.
An attempt is made to show the topological transition on conformal manifold for quantum Ising model with longer range interaction. This model Hamiltonian system has different gapped phases with different topological indices and also different quantum critical lines depending on the presence and absence of the transverse field. We also present the central charge for the different regimes of the parameter space. We show explicitly on the interplay of criticality, topology and central charge (C) in the presence and absence of transverse field and also non-universal character of C. We show how the Lifshitz transition occurs in the presence of transverse field and changes the universality class of the central charge in presence of transverse field. We show explicitly the existence of conformal field theory (CFT) criticality and non-CFT criticality. We have presented an explicit calculations to find the relation between the polynomial function and the Anderson-pseudo spin model Hamiltonian. Our results are far more enrich than the existence results of noninteracting fermionic many body system. This work not only provides a new perspective in topological state of conformal field theory but also for low dimensional quantum many body system.
We study the topology and edge mode physics at quantum criticality for correlated quantum matter which was introduced in Nature Physics 1.1, 53(2005). Our model Hamiltonian belongs to BDI symmetry class with short range and long range interactions. We observe an interesting behavior that edge modes behave differently although the Hamiltonian belongs to same symmetry class. We do the complex analysis to unveil the topological properties of the Hamiltonian. We show, the physics of the edge mode is not the same for all quantum critical lines since we find gapless edge modes at criticality. We also show, merging of zero modes with the bulk for certain parameter regime. The central charge (c) are used to characterize different phases of the model Hamiltonian.
Topology is a fascinating phenomena in condensed-matter physics typically associated with a bulk gap. However, recent research shifts focus to quantum critical points or phases that exhibit nontrivial topological properties. Here we explore a cluster-Ising chain with long-range antiferromagnetic interactions that decay as a power law with distance. Using large-scale density matrix renormalization group simulations, we demonstrate that the nontrivial topology at the critical point remains stable against long-range interactions, resulting in a topologically nontrivial critical line. Moreover, even within the gapped region, the interplay between topology and long-range interaction can give rise to a topological phase featuring algebraically decaying correlations and edge modes, similar to gapless topological phases. We refer to this phase as the algebraic topological phase, which exhibits nontrivial gapless topological behaviors and arises solely from long-range interactions without short-range counterparts. The findings pave the way for more studies on topological states in long-range systems. Long-range interactions play a crucial role in condensed-matter systems and can lead to range of emerging and exotic physical phenomenon. Here using simulations, the authors investigate the non-trivial topological properties that can arise in a cluster Ising chain with long-range antiferromagnetic interactions, finding evidence for what they classify as an algebraic symmetry-protected topological phase.
An attempt is made to study the quantum criticality in non-Hermitian system with topological characterization. We use the zero mode solutions to characterize the topological phases and, criticality and also to construct the phase diagram. The Hermitian counterpart of the model Hamiltonian possess quite a few interesting features such as Majorana zero modes (MZMs) at criticality, unique topological phase transition on the critical line and hence these unique features are of an interest to study in the non-Hermitian case also. We observe a unique behavior of critical lines in presence of non-Hermiticity. We study the topological phase transitions in the non-Hermitian case using parametric curves which also reveal the gap closing point through exceptional points. We study bulk and edge properties of the system where at the edge, the stability dependence behavior of MZMs at criticality is studied and at the bulk we study the effect of non-Hermiticity on the topological phases by investigating the behavior of the critical lines. The study of non-Hermiticity on the critical lines revels the rate of receding of the topological phases with respect to the increase in the value of non-Hermiticity. This work gives a new perspective on topological quantum criticality in non-Hermitian quantum system.
In an attempt to theoretically investigate the quantum phase transition and criticality in topological models, we study a Kitaev chain with longer-range couplings (finite number of neighbors) as well as truly long-range couplings (infinite number of neighbors). We carry out an extensive topological characterization of the momentum space to explore the possibility of obtaining higher order winding numbers and analyze the nature of their stability in the model. The occurrences of phase transitions from even-to-even and odd-to-odd winding numbers are observed with decreasing longer-rangeness in the system. We derive topological quantum critical lines and study them to understand the behavior of criticality. A suppression of higher order winding numbers is observed with decreasing longer-rangeness in the model. We show that the mechanism behind such phenomena is due to the superposition and vanishing of the topological quantum critical lines associated with the higher winding number. Through the study of the Berry connection, we show the possible different behaviors of critical lines when they undergo superposition along with the corresponding critical exponents. We analyze the behavior of the long-range models through the momentum space characterization. We also provide the exact solution for the problem and discuss the experimental aspects of the work.